The skewrank of oriented graphs
Abstract
An oriented graph $G^\sigma$ is a digraph without loops and multiple arcs, where $G$ is called the underlying graph of $G^\sigma$. Let $S(G^\sigma)$ denote the skewadjacency matrix of $G^\sigma$. The rank of the skewadjacency matrix of $G^\sigma$ is called the {\it skewrank} of $G^\sigma$, denoted by $sr(G^\sigma)$. The skewadjacency matrix of an oriented graph is skew symmetric and the skewrank is even. In this paper we consider the skewrank of simple oriented graphs. Firstly we give some preliminary results about the skewrank. Secondly we characterize the oriented graphs with skewrank 2 and characterize the oriented graphs with pendant vertices which attain the skewrank 4. As a consequence, we list the oriented unicyclic graphs, the oriented bicyclic graphs with pendant vertices which attain the skewrank 4. Moreover, we determine the skewrank of oriented unicyclic graphs of order $n$ with girth $k$ in terms of matching number. We investigate the minimum value of the skewrank among oriented unicyclic graphs of order $n$ with girth $k$ and characterize oriented unicyclic graphs attaining the minimum value. In addition, we consider oriented unicyclic graphs whose skewadjacency matrices are nonsingular.
 Publication:

arXiv eprints
 Pub Date:
 April 2014
 arXiv:
 arXiv:1404.7230
 Bibcode:
 2014arXiv1404.7230L
 Keywords:

 Mathematics  Combinatorics;
 05C20;
 05C50;
 05C75
 EPrint:
 17 pages, 4 figures